Functional Analysis

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- ]. (2-3) Riemann sphere 29 According to our glossary, these properties mean to = f (z) is defined on the z sphere with values on the to sphere.

Now (2-15) may seem strange because of the following example: The function f(z) = 1/z is analytic at z = a, [since f(1/z) = z is analytic at z = 0], but by (2-15), f (z) has a residue -1 at z = oo. The residue, therefore, was not defined as a property of f (z) but a property of the differential f (z) dz. Let the following be the Laurent expansion for large z: (2-16) f(z) _ Cnzn = c + zll +. Riemann sphere (2-5) 33 Then to find the residue at - for f(z), we could consider f(1/zz)dz f(1) d(1)= J (2-17) z Then in (2-16) we obtain "differential" St, formally written as (2-18) - c_1 dz z C-1 Co z zz _ Cl z3 1dz + irrelevant terms Thus, indeed, the residue at oo of f(z) in (2-16) is -c_l.

MOBIUS STRIP Consider the domain S given by coordinates (in the xy plane) (1-23) 0

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